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We use network analysis to describe and characterize an archetypal quantumsystem - an Ising spin chain in a transverse magnetic field. We analyzeweighted networks for this quantum system, with link weights given by variousmeasures of spin-spin correlations such as the von Neumann and Renyi mutualinformation, concurrence, and negativity. We analytically calculate thespin-spin correlations in the system at an arbitrary temperature by mapping theIsing spin chain to fermions, as well as numerically calculate the correlationsin the ground state using matrix product state methods, and then analyze theresulting networks using a variety of network measures. We demonstrate that thenetwork measures show some traits of complex networks already in this spinchain, arguably the simplest quantum many-body system. The network measuresgive insight into the phase diagram not easily captured by more typicalquantities, such as the order parameter or correlation length. For example, thenetwork structure varies with transverse field and temperature, and thestructure in the quantum critical fan is different from the ordered anddisordered phases.

A complex network description of thermal quantum states in the Ising spin chain